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Parallel planes are two planes in three-dimensional space that never intersect. They remain at a constant distance from each other. Often, they have normal vectors that are scalar multiples of each other. When you need to determine if two planes are parallel, checking their normal vectors is crucial.
For a plane described by the equation:

• - a normal vector can be derived from the coefficients of its terms.
• If the normal vectors of two planes are parallel, the planes are parallel as well.

In our example, we are asked to find planes that are parallel to the coordinate planes: the xy-plane, yz-plane, and xz-plane.

• The xy-plane's equation can be written as \( z = 0 \), with a normal vector of \((0, 0, 1)\).
• The yz-plane has a normal vector \((1, 0, 0)\) corresponding to \( x = 0 \).
• Similarly, the xz-plane has a normal vector \((0, 1, 0)\), or \( y = 0 \).

When constructing a plane parallel to these, maintain the same form but adjust the equation to pass through the given point.

Coordinate planes are fundamental in three-dimensional geometry. These planes correspond to the axes: xy-plane, yz-plane, and xz-plane. Each serves as a benchmark for positioning other planes. The xy-plane, represented by \( z = 0 \), contains all points for which the z-coordinate is zero. For instance, in the task, a plane parallel to this will have the equation \( z = c \), here \( z = 1 \), since it needs to be parallel and pass through the point \((3, -1, 1)\).The yz-plane is characterized by \( x = 0 \), where all x-coordinates are zero. A plane parallel to this will have the equation \( x = c \). Here, it is \( x = 3 \).Lastly, the xz-plane is defined as \( y = 0 \), and for our purposes, a parallel plane would be expressed as \( y = c \), given in the task as \( y = -1 \).These planes are important reference tools. They help in visualizing and formulating equations for parallel planes in space.

Vectors are mathematical objects that have both magnitude and direction. They are crucial for analyzing planes. Each vector can be broken down into components corresponding to particular directions, often aligned with the coordinate axes.In expressing planes, vectors help in finding normals. These are perpendicular vectors that describe the plane's orientation. A normal vector to a plane is integral to its equation. For example,

• The vector \((0, 0, 1)\) indicates the plane's normal is purely in the z-direction.
• This implies the plane itself extends infinitely in the x and y directions.

When dealing with parallel planes, their normal vectors share proportional components. To form equations of parallel planes, use components from the initial plane's normal vector. If a plane is parallel to the xy-plane, its normal vector will be similar to that of the xy-plane itself.Understanding vector components aids greatly in comprehending the orientation and behavior of planes in 3D geometry. This, in turn, assists in crafting equations for planes that perfectly match the required conditions.